A family of quadratic scoring rules (QSR's) is defined. Some properties of these scoring rules are described, and it is demonstrated that QSR's are strictly proper. The probability (or Brier) score and the ranked probability score are shown to be special cases of the general QSR.
A geometrical framework for the representation of QSR's is presented. This framework facilitates formulation of QSR's and provides insight into the properties of these scoring rules, including the sensitive-to-distance property. The relationships between QSR's and measures of the value of (probability) forecasts are briefly discussed.
The richness of the family of QSR's provides the evaluator with considerable flexibility in choosing a scoring rule that is particularly suited to the situation at hand.